Many products are packaged in bottles or other containers whose mouths are sealed by a thin membrane of a suitable material, such as a metallic foil or a plastic film, until a consumer or other user is ready to remove the contents of the container for consumption or use after piercing of the sealing membrane. Various types of infant formula are packaged in this manner. In addition to the membrane that seals the mouth of such a container, it is also customary to apply a removable molded plastic closure to the container to protect the membrane from damage during shipment and storage of the filled container and to re-close the container after partial removal of its contents, the membrane no longer serving to seal the container at this time due to its piercing. Typically, in such a packaging application the piercing of the membrane is accomplished by one or more piercing elements on the top of the closure after removing the closure and reapplying it to the container in an inverted orientation. U.S. Pat. No. 5,709,311 (Butler et al.), U.S. Pat. No. 5,505,326 (Junko), and U.S. Pat. No. 5,090,582 (Art et al.) describe closures of this general type.
Piercing closures of the type described are also used to cut the membrane through an arc of substantial extent, often 360.degree., by turning the closure on the container, after inverting the closure on the container. Unfortunately, the quality of this cutting action often degrades as the length of the cut increases, due to a loss in the tension in the membrane as the cutting action proceeds. This problem makes it difficult to properly remove the contents of the container, with resultant aggravation of the user or consumer and loss of good will of the manufacturer whose product is packaged in the container. Further, it is quite desirable that the cutting of the membrane which seals the mouth of the container not extend over an arc of 360.degree., so that the severed membrane will remain attached to the container and not drop into its contents